A group of 12 countries is considering whether to form a monetary union. Theydif

Question

A group of 12 countries is considering whether to form a monetary union. Theydiffer in their assessments of the costs and benefits of this move, but each standsto gain more from joining, and lose more from staying out, when more of theother countries choose to join. The countries are ranked in order of their liking forjoining, 1 having the highest preference for joining and 12 the least. Each countryhas two actions, IN and OUT. Let

B(i,n) = 2.2 + n i

be the payoff to country with ranking i when it chooses IN and n others havechosen IN. Let

S(i,n) = i n

be the payoff to country with ranking i when it chooses OUT and n others havechosen IN.

(a) Show that for country 1, IN is the dominant strategy.

(b) Having eliminated OUT for country 1, show that IN becomes the dominantstrategy for country 2.

(c) Continuing in this way, show that all countries will choose IN.

(d) Contrast the payoffs in this outcome with those where all choose OUT. Howmany countries are made worse off by the formation of the union?

Explanation / Answer

a) For country 1 rank =1 so i=1

Now the payoff for choosing IN B1 = 2.2+n-1 = 1.2+n where n could be 0 to 11 so worst case scenerio is n=0 so minimum payoff for B1 = 1.2

Now the payoff for choosing OUT S1 = 1-n where n could be 0 to 11 so best case scenerio is n=0 so maximum payoff for S1 = 1. So as we see that minimum payoff for B1 is more than maximum payoff for S1 so country 1 will always choose IN.

b) Now we know that atleast 1 country wil choose IN

For country 2 rank =2 so i=2

Now the payoff for choosing IN B2 = 2.2+n-2 = 0.2+n where n could be 1 to 11 so worst case scenerio is n=1 so minimum payoff for B2 = 1.2

Now the payoff for choosing OUT S2 = 2-n where n could be 1 to 11 so best case scenerio is n=1 so maximum payoff for S2 = 1. So as we see that minimum payoff for B2 is more than maximum payoff for S2 so country 2 will always choose IN.

c) Now we know that atleast 2 country wil choose IN

For country 3 rank =3 so i=3

Now the payoff for choosing IN B3 = 2.2+n-3 = -0.8+n where n could be 2 to 11 so worst case scenerio is n=2 so minimum payoff for B3 = 1.2

Now the payoff for choosing OUT S3 = 3-n where n could be 2 to 11 so best case scenerio is n=2 so maximum payoff for S3 = 1. So as we see that minimum payoff for B3 is more than maximum payoff for S3 so country 3 will always choose IN.

Now we know that atleast 3 country wil choose IN

For country 4 rank =4 so i=4

Now the payoff for choosing IN B4 = 2.2+n-4 = -1.8+n where n could be 3 to 11 so worst case scenerio is n=3 so minimum payoff for B4 = 1.2

Now the payoff for choosing OUT S4 = 4-n where n could be 3 to 11 so best case scenerio is n=3 so maximum payoff for S4 = 1. So as we see that minimum payoff for B4 is more than maximum payoff for S4 so country 4 will always choose IN.

So we see that for all countries knowing that higher ranked countries have chosen IN then their minimum payoff for IN is more than maximum payoff for OUT so they will chose IN.

4) As seen when all choose out then n=0

So only first country is better off by formation of union.

Country in = 2.2-i out i 1 1.2 1 2 0.2 2 3 -0.8 3 4 -1.8 4 5 -2.8 5 6 -3.8 6 7 -4.8 7 8 -5.8 8 9 -6.8 9 10 -7.8 10 11 -8.8 11 12 -9.8 12

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